Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation.

*(English)*Zbl 0872.34033The aim is to investigate the structure of the homoclinic and heteroclinic solutions of the scalar fourth order equation

$$\gamma {u}^{\left(4\right)}-\beta {u}^{\text{'}\text{'}}+{F}^{\text{'}}\left(u\right)=0,$$

with $\gamma ,\beta >0$. The two primary examples

$${F}_{1}\left(u\right)={\textstyle \frac{1}{4}}{({u}^{2}-1)}^{2}\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{F}_{2}\left(u\right)={\textstyle \frac{2}{{\pi}^{2}}}(1+cos\pi u)$$

are considered as a background of the theory. The authors construct a countable family of multitransition homoclinic and heteroclinic solutions. The glueing method applied in the paper can also be used to obtain periodic orbits which are in an arbitrarily small neighborhood of the heteroclinic loop and are structurally similar to those which would be found using dynamical methods of Devaney. Also, the existence of the family of homoclinic and heteroclinic solutions described is sufficient to show that the dynamics are chaotic.

Reviewer: Ding Tongren (Beijing)

##### MSC:

34C37 | Homoclinic and heteroclinic solutions of ODE |